论文标题
多扭码代码的双面galois二元
The two-sided Galois duals of multi-twisted codes
论文作者
论文摘要
近几十年来，表征具有丰富代数结构的线性代码的双重代码引起了极大的兴趣。开始是将有限场上的循环代码表示为多项式环中的理想。随后，研究了康斯坦西克莱克，准循环，准串联，普遍的准循环和多扭曲的代码的双重，在文献中广泛出现。我们考虑了多键（MT）代码的类别，因为它扩展到所有这些代码。我们将MT代码$ \ Mathcal {C} $描述为主要理想域上的模块。因此，$ \ MATHCAL {C} $具有满足相同方程的生成器多项式矩阵（GPM）。 $ \ Mathcal {C} $的降低的GPM是其GPM的Hermite正常形式。我们表明欧几里得双$ \ Mathcal {C}^\ perp $ $ \ Mathcal {C} $也是MT。我们使用$ \ Mathcal {C} $的$ \ Mathcal {C}^\ perp $的GPM的公式证明了一个公式。然后，我们的目标是用Galois Dual代替欧几里得双重。 Galois内部产物是一种不对称形式，因此我们区分右侧和左Galois双重。我们表明，MT代码的左右与左GALOIS双重二线也是MT，但可能具有不同的变化常数。我们的研究是第一个同时包含线性代码的右和左Galois二重要的研究。这给出了两个优点：建立其相互联系的身份，并引入以前从未出现在文献中的双面Galois双重偶。我们使用MT代码的双面GALOIS双重双向dual使用条件，因此将其GPM表征为特征。还研究了两个特殊情况，一个是当左右galois双人双重相交时，另一个是当它们重合时。对于任何线性代码，都考虑了后一种情况，在该法规中为左右与二元组的平等建立了必要和充分的条件。
Characterizing the duals of linear codes with rich algebraic structures received great interest in recent decades. The beginning was by representing cyclic codes over finite fields as ideals in the polynomial ring. Subsequently, studying the duals of constacyclic, quasi-cyclic, quasi-twisted, generalized quasi-cyclic, and multi-twisted codes appeared extensively in literature. We consider the class of multi-twisted (MT) codes because it extends to all of these codes. We describe a MT code $\mathcal{C}$ as a module over a principal ideal domain. Hence, $\mathcal{C}$ has a generator polynomial matrix (GPM) that satisfies an identical equation. The reduced GPM of $\mathcal{C}$ is the Hermite normal form of its GPM. We show that the Euclidean dual $\mathcal{C}^\perp$ of $\mathcal{C}$ is MT as well. We prove a formula for a GPM of $\mathcal{C}^\perp$ using the identical equation of the reduced GPM of $\mathcal{C}$. Then we aim to replace the Euclidean dual with the Galois dual. The Galois inner product is an asymmetric form, so we distinguish between the right and left Galois duals. We show that the right and left Galois duals of a MT code are MT as well but with possibly different shift constants. Our study is the first to contain the right and left Galois duals of a linear code simultaneously. This gives two advantages: establishing their interconnected identities and introducing the two-sided Galois dual that has not previously appeared in the literature. We use a condition for the two-sided Galois dual of a MT code to be MT, hence its GPM is characterized. Two special cases are also studied, one when the right and left Galois duals trivially intersect and the other when they coincide. The latter case is considered for any linear code, where a necessary and sufficient condition is established for the equality of the right and left Galois duals.
